C1 chapter 3 - equations and inequalities

"In this chapter you will learn how to simultaneous equations and solve linear and quadratic inequalities."

Starting point: RISP 34

On the way, we'll need to consolidate our understanding of linear, reciprocal and quadratic functions. We'll do some solving of linear and quadratic equations and we'll need to be fluent at sketching a range of graphs.
A nice introduction to that comes from Jonny Griffiths' RISP 34. Here's the idea:
Suppose you have some small cards with the symbols:
Here's a worksheet to generate the cards
How many distinct, meaningful equations can you make with some or all of them? (No repeating symbols and no exponents for now, please)
What do the graphs of these look like as you vary the constant term c ?

Enter your equations into some graphing software.
Use the value c = 1 to start with.
Now alter c . Use a slider if you can.

Are there any lines of symmetry? Are they invariant as you change c ?

Can you find a value for c so that:
Three of your curves are straight lines that enclose an equilateral triangle? What is its area?
Two of your equations represent a curve and a straight line that touch? Where do they touch?
What other questions could we ask about this situation?
[scroll down to reveal the possible graphs - which ones did you find?]







Download this as a GeoGebra file:
There are equilateral triangles possilble like these:
and
How might you go about finding the exact values for c in each case?
You can get a line to just touch the curve when:
or
Again, how can you be certain of the exact value of c
Extension: what if we allow any or all of x, y, c, to be squared? How many equations now?

Sections

C1 section 3.1

L.O: WALT solve simultaneous linear equations by elimination.
This means that you manipulate one or both linear equations in such a way that by adding or subtracting them, you can eliminate one of the variables. (You can make one of the letters 'disappear.')
Here are some examples:

Example 3.1.1.
Solve the simultaneous equations:
$x + y = 2$
$y - x = 2$
Worked solution:

This works neatly because there is an x in equation A and -x in equation B. This means that when we add the equations, the x is eliminated.
Notice also that this corresponds to two of the graphs in the GeoGebra tool above where c = 2:
$x + y = 2$
and
$y = x$
Here's a screenshot:
What does your solution
$x = 1, y = 1$
tell you about the graph?

Example 3.1.2:
But what if it wasn't so neat?
Solve the simultaneous equations:
$2x + y = 6$
$x + 2y = 6$
Worked solution:

Do you notice that the decision to make the y values match was arbitrary? We could have chosen to make the x values match instead:

But of course, we'll get the same solution.
Here is a GeoGebra graph tool that corresponds to this type of problem. You can change the equations if you like:







Hopefully I've set the defaults to match the next example:
Example 3.1.3:
Solve the simultaneous equations
$2x + 3y = 5$
$3x + 2y = 10$
You know what solution to expect from the graph don't you?

Intersection at (4,-1)

Can you prove that by using algebra?
Here's a worked solution:

Again the decision to eliminate the y variable first was arbitrary. You could have multiplied A by 3 and B by 2 to allow you to eliminate x instead. Either way, the key thing is to look at the coefficients of x and find their LCM and do the same for the coefficients of y. This will help you decide which variable is easiest to eliminate and what to multiply each equation by to get them to cancel.
You need to know:


Examples:

C1 Exercise 3A

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• C1 Exercise 3A worked solutions:
• 

  C1 Exercise 3A (simultaneous equations).jnt
  

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  C1 Exercise 3A (simultaneous equations).pdf
  

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  • 269 KB
• 

  C1 section 3.1 - simultaneous equations by elimination.jnt
  

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  • 85 KB
  

  C1 section 3.1 - simultaneous equations by elimination.pdf
  

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• Questions similar to this exercise:

C1 section 3.2

You need to be able to: solving simultaneous linear equations by substitution

You need to know:

Examples:

C1 Exercise 3B

• C1 Exercise 3B worked solutions:
  

  C1 Exercise 3B (simultaneous eqns Substitution).jnt
  

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  • 77 KB
• Questions similar to this exercise:

C1 section 3.3

You need to be able to: using substitution to solve a pair of simultaneous equations where one is linear and the other is quadratic.

You need to know:

Examples:

C1 Exercise 3C

• C1 Exercise 3C worked solutions:
  

  C1 Exercise 3C (simultaneous eqns with non-linear fns).jnt
  

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• You may wish to look at these visualisations of the first two questions to see why they make sense:
• 

  C1 Exercise 3C visualisation 1a.jnt
  

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  and
  

  C1 Exercise 3C visualisation 1b.jnt
  

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  • 39 KB
• Questions similar to this exercise:

C1 section 3.4

You need to be able to: solve linear inequalities.

You need to know:

Examples:

C1 Exercise 3D

• C1 Exercise 3D worked solutions:
• Questions similar to this exercise:

C1 section 3.5

You need to be able to: solve quadratic inequalities.

You need to know:

Examples:

C1 Exercise 3E

• C1 Exercise 3E worked solutions:

C1 Exercise 3E question 3a looks like this:

I set out the steps to this solution on my GeoGebra solution page for C1 Ex3E Q3a

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  C1 Exercise 3E question 4 (appears only in new version of book)

• Questions similar to this exercise:

C1 Exercise 3F

• C1 Exercise 3F (miscellaneous exercise) worked solutions: